Adding Fractions

This is the 4th in a series of posts fleshing out the material written by Hung Hsi Wu in Critical Concepts for Understanding Fractions. See also Part I ,Part II, and Part III.

We left off last time with an introduction to common denominators. First, we showed that a given number is unique and represented by a unique location on the number line, even if it has several different representations. For fractions, these different representations are often made manifest by representing the number in terms of different denominators. For example,

Next, we showed that it was always possible to represent two fractions, originally with two different denominators, as fractions with the same denominator. That is:

if m/n and k/l are fractions (where n and l are nonzero), then m/n = ml/nl and k/l = nk/nl. That is, both now share the denominator nl.

For example, take 4/3 and 1/5. Both can share the common denominator 15: 4*5/3*5 = 20/15. 3*1/3*5 = 3/15.



Thus, we are now able to talk about these two fractions as having a common denominator. Once they are represented with common denominators, it becomes easy to compare fractions. Specifically, a/nl is less than b/nl if a < b.

We now move on to how common denominators can help us add fractions.

Fundamentally, adding fractions is very similar to adding whole numbers on the number line. Recall that we could add 4 and 3 by making successive unit jumps on the number line: starting at zero, jumping 4 unit steps to the right, and then jumping 3 more units resulted in 7. Recall that a fraction m/n is defined as the location on the number line after partitioning the unit segment into n partitions, and then moving m jumps to the right of zero. Similarly, the fraction m/n could be represented as a segment whose left marker began at 0 and right endpoint was at the location m/n. Further, it could be represented by the length m/n--this length being the same no matter where on the number line it is located. The addition of fractions k/l + m/n is the concatenation of two segments: the first segment whose length is k/l, and a second segment whose length is m/n.

This leads us to recognize that whole numbers are a special form of fractions: a whole number k is just the fraction k/1, where the unit segment is left as one segment, and we then make k jumps to the right on unit segments. So a whole number is special case of a fraction, a fraction whose denominator is 1. And more importantly, the addition of whole numbers is really the addition of fractions with common denominators of 1.


Now, we start to think about the addition of fractions as we thought of the addition of whole numbers:

To add k/l + m/n, we first partition each unit segment of the number line into l pieces, and then concatenate k segments to the right. Next, we create a new set of unit segments, whose origin is k/l, and then partition each unit segment of this number line into n pieces, and concatenate m segments to the right. Here's an example of the sum of 2/3 and 3/4:



This method does, in fact, find the unique location on the number line that represents this fraction. That is, we've properly added the fractions and found the new location. But while this finds the unique location on the number line that represents this fraction, this location doesn't really help us to simplify our representation: we'd like to represent this new fraction in the standard form of a/b. But that means that in the end, we'd like our new fraction to be easily represented by just one set of partitions: the bths. If we could find a way to draw both k/l and m/n on the same partitions, we would know how many total jumps we've made on one set of partitions, rather than on these two different partitions.

To do this, we need to rewrite k/l and m/n with common denominators.

To do this, we rewrite k/l as kn/ln, m/n as lm/ln, and now we can rewrite k/l + m/nas
k/l + m/n = kn/ln + lm/ln = (kn + lm)/ ln.

Applying this to our above number line, this says first, we partition each unit segment into ln pieces. Then we make kn jumps to the right, followed by lm jumps to the right.
In the case of 2/3 and 3/4, we rewrite 2/3 as 2*4/3* = 8/12, and we rewrite 3/4 as
3*3/3*4 = 9/12.



The sum, then of 2/3 and 3/4 is 2*4/3*4 + 3*3/3*4 = 8/12 + 9/12 = 17/12.


Is it possible to show the 2/3 + 3/4 to an average 5th grader without confusing them more? Does showing how the unit segments don't line up help motivate why we are finding a new common denominator? Or does it just leave them wondering how in the world adding fractions is meaningful? From my perspective, it seems important to get them to see that the location on the number line is the same, but our new hash marks help us to write a simpler number. Fundamentally, we're saying that 2/3 + 34 IS 2/3 + 3/4. I would hope questions on homework would be phrased as "simplify 2/3 + 3/4" rather than "solve 2/3 + 3/4". We're not solving an equation. We're finding an equivalent expression. I think when showing the students, the point to stress is that we want to add the fractions and turn the sum into our standard form where we have a single set of partitions.

Next, we'll discuss mixed numerals and summing before moving on to subtraction.

Carolyn on Connected Muck

email from Carolyn:

Grr. We just did a Connected Muck homework project on box-and-whisker plots. The kid is supposed to look at a huge sheet of data on the length of arrowheads from several different sites (each having maybe 40 arrowheads), then derive box plots from each one of them, then derive two more box plots from new sites and compare them. Then he is supposed to do the same thing all over again for the widths of all the arrowheads, and this time the widths aren’t even in an ordered list! You’d have to be obsessive-compulsive to be willing to do all that by hand.

Do you remember how people were always talking about how adults who were taught math the old-fashioned way couldn’t understand the CMP homework? We-he-hell. I understand exactly what they’re getting after in every one of their assignments; the trouble is the goal is STUPID. What they’re usually getting after is making some connection to calculus or higher math that the kids aren’t ready to take in yet, or they are trying to get kids to think flexibly about a topic they have just then been exposed to, and haven’t learned much less mastered yet. It’s stupid. It flies in the face of cognitive science studies.

3-11-2008

In person, Carolyn always says "We-he-hell." I can hear her saying it now!

We need to beseech Carolyn to come back and write beaucoup posts on Connected Muck and sundry.

from the ice cream shop

more from Anonymous Engineer:

One year I had to sit through endless hours of meetings, analyzing test results (Massachusetts MCAS). The principal, VP, and 10 or so teachers would sit around a table and look for anomolous results (pretty easy to find) where our school really sucked vs. the state average.

Then we would basically be asked to guess what went wrong. We had hundreds of square feet of 'what went wrong' guesses posted all over the conference room. One particular question (with a horrible failure rate) has always stuck with me. It was something like "How many quarter pound hamburgers can you make from 3 pounds of hamburg?"

Our guess? Kids are having problems dividing whole numbers by fractions. I wasn't comfortable with the guess since it was my class that bombed this question and I was very confident that my kids knew how to do such a simple problem. As luck would have it, one of my really bright students was passing the room and I called her in to ask her about the question.

She floored us all. "You can't make hamburg from hamburg." Huh? It seems that in Puerto Rico (school is > 85% Hispanic population) the source material for hamburg is called ground meat. In our New England cacoon we make hamburger patties from hamburg. The real story is that kids were totally baffled by the seemingly bizarre question.

Please don't think I've told this little story to push on test bias. That could be a whole 'nother blog. No, I bring it up as an example of the futility of analyzing test results, especially when there's no kids in the room. You really have no idea what went wrong when kids bomb a question.

Regards from the ice cream shop.

Meanwhile, I'm sitting here thinking, "Hamburg"?

What is "hamburg"?

In the Midwest, we make hamburger out of hamburger. Somehow I managed to live in Massachusetts for 3 years of my life and never pick up on the fact that, apparently, folks in Massachusetts make hamburger out of hamburg.

From time to time I try to think exactly what kind of book could or should be spun off from Kitchen Table Math.

I'm pretty sure it's a book of stories from inside the black box.

Or the ice cream shop, as the case may be.

Zig's New Book: Chapter 5

"Millions of needy kids have been robbed of career ladders by the Feds' decisions. Millions of teachers have been professionally insulted because information about DI practices never reached them or the unfortunate educators who trained them." - Siegfried "Zig" Engelmann

The fifth chapter of Engelmann's most recent book, Teaching Needy Kids in Our Backward System, is posted at zigsite.

Chapter 5 discusses the evaluation of Project Follow Through. Most educators are not familiar with it, and it is all but unknown outside the educational community. That is ironic because Follow Through was the largest educational experiment ever conducted. It involved over 200,000 students and 22 sponsors of different approaches for how to teach at-risk children in grades kindergarten through 3. The 178 communities that implemented the different approaches spanned the full range of demographic variables (geographic distribution and community size), ethnic composition (white, black, Hispanic, Native American) and poverty level (economically disadvantaged and economically advantaged).

[snip]

The project began in 1968 and was evaluated nine years later, after the various sponsors had a reasonable time period to de-bug their operation. Direct Instruction was one of the sponsors. Chapter 5 picks up the story from there.

While Teaching Needy Kids has had me shaking my head in disbelief and frustrated to the point of having to put the book down for a moment to take a deep breath, it records a significant moment in educational history. It should be read by every person involved in education be they parents, teachers, or administrators if only because it is the untold story that deserves to be heard. If we can learn from our past mistakes and find a way to repair them, all the better.

Teaching Needy Kids in Our Backward System
Siegfried "Zig" Englemann
Chapter 5
Follow Through Evaluation

What Can Cookies Do For Students?

A couple days ago, the NYT carried an interesting Op-Ed written by a professor of molecular biology and neuroscience and an editor of Nature Neuroscience on how the brain responds to tasks requiring "willpower." The article, Tighten Your Belt, Strengthen Your Mind, was ostensibly discussing the mortgage/economic crises that some families might be facing; but there are some fascinating observations made in the article that could improve teaching and learning.

The brain has a limited capacity for self-regulation, so exerting willpower in one area often leads to backsliding in others. . . . The brain’s store of willpower is depleted when people control their thoughts, feelings or impulses, or when they modify their behavior in pursuit of goals. Psychologist Roy Baumeister and others have found that people who successfully accomplish one task requiring self-control are less persistent on a second, seemingly unrelated task.

In one pioneering study, some people were asked to eat radishes while others received freshly baked chocolate chip cookies before trying to solve an impossible puzzle. The radish-eaters abandoned the puzzle in eight minutes on average, working less than half as long as people who got cookies or those who were excused from eating radishes. Similarly, people who were asked to circle every “e” on a page of text then showed less persistence in watching a video of an unchanging table and wall.

First thought -- anytime chocolate is involved, it's got to be a good thing. We need more research like this.

But seriously, eating cookies improves your ability to stick to a difficult or boring task. Who else is thinking about automatic recall of math facts? Perhaps all we need to do to hit our national goals is give kids cookies instead of radishes.

Last month when the CMT tests were being administered to all those 3rd through 8th graders, parents were asked to donate boxes of healthy snacks for the classrooms -- granola bars, fresh fruit, carrot sticks, etc. Perhaps we are sealing our own fate with this behavior -- if we'd have sent chocolate chip cookies and tossed the carrot sticks, maybe our kids could all hit goal.

But things get even better as we read through the op-ed:

What limits willpower? Some have suggested that it is blood sugar, which brain cells use as their main energy source and cannot do without for even a few minutes. Most cognitive functions are unaffected by minor blood sugar fluctuations over the course of a day, but planning and self-control are sensitive to such small changes. Exerting self-control lowers blood sugar, which reduces the capacity for further self-control. People who drink a glass of lemonade between completing one task requiring self-control and beginning a second one perform equally well on both tasks, while people who drink sugarless diet lemonade make more errors on the second task than on the first. Foods that persistently elevate blood sugar, like those containing protein or complex carbohydrates, might enhance willpower for longer periods.

If you need to study for a big exam, it might be smart to let the housecleaning slide to conserve your willpower for the more important job. Similarly, it can be counterproductive to work toward multiple goals at the same time if your willpower cannot cover all the efforts that are required. Concentrating your effort on one or at most a few goals at a time increases the odds of success.

Clearly, we all need to stop insisting our kids clean their rooms and clear their dishes if we expect them to sit down and do their homework or study for an exam.

But what does this say about our schools? Minutes off of recess is so common a punishment its not even worth noting. Carrot sticks and fruit for birthday snacks? No rough-housing during recess, when you get recess? Are we expecting our kids to have far more willpower to sit and do dull tasks than is reasonable given the human limitations of willpower? Perhaps schools should rethink the school day in terms of willpower depletion -- classroom tasks can be less engaging (and more productive) if kids are given a freer rein, or more sugar, at other times of the day.

I'm hoping Catherine will jump in here, because she has so much more cognitive studies at her fingertips than I ever will.

Mark Roulo & Steve H on open vs closed loops

An open loop control system is one in which you don't bother checking to see how you are doing. An example would be a robot arm, where to move to a certain position, one just moves a motor a certain number of revolutions, but doesn't check to see if maybe the gears are slipping.

Open loop systems tend to be simpler and cheaper than closed loop systems, but less reliable.

A closed loop system would be the robot arm with a sensor that fed back the arm position. In this case, even with slightly slipping gears you can get the arm to the correct location because the robot might issue one extra revolution command to make up for the slipping gears.

Poor schools and classrooms tend to run open-loop ... no one knows if the kids are learning what they are supposed to learn. Well run schools and classrooms run closed-loop by checking the kids for progress AND THEN TAKING ACTION (e.g. reteaching). Even better would be a school or classroom that was closed loop on the teaching itself. For example, if kids kept not learning, trying to figure out what to change so that the kids learned the first time around.

I don't get the impression that the second version is done formally in very many schools (or businesses for that matter).

Positive feedback means you do more as you get closer to what you want. I don't think any engineers design these things on purpose, but sometimes you get one anyway (I think ram-jets have a built in positive feedback ... without external controls, the faster they go the faster they want to go until they burn themselves up).

Mark Roulo

Steve H:

"I don't get the impression that the second version is done formally in very many schools.."

Because of NCLB, our state mandates that this process takes place. They send out (almost worthless) questionnaires to parents and students and they hold parent-teacher meetings to review the standardized test results.

For example, the committee might see that the standardized test scores for problem solving in math have gone down a few points. They will sit around and discuss the problem. It's down only a slight bit, so there is not much of a problem, right? Perhaps it's due to the small sample size (they get to sound rigorous). Well, in any case, the administration will tell the teachers in their next meeting to emphasize problem solving more. OK, next problem.

The flaw, as you can see, is that this is all relative. You can provide all of the feedback you want, but if the basic structure of the system is wrong, it will never be fixed. Standardized tests will never tell you that many more kids should be getting to algebra in 8th grade. In fact, it tells the school and many parents the opposite. (They conveniently ignore international tests.)

This reminds me of my son's Science Olympiad project. He can tweak things all he wants, but his basic vehicle design will never reach the required maximum distance.

In our district, "Extra Help" is the answer to all learning difficulties, across the board. Students are supposed to "be proactive and seek extra help." If a student does not seek Extra Help then he deserves his C or D or F. At the high school, students are to seek extra help and also practice "self-advocacy," which means parents are not to contact the teacher or the administration about a problem until the student has tried and failed to solve the problem on his or her own.

This philosophy has now been imported into the 4-5 school. When parents raise an issue with a teacher, the parent is told that the student should have come to the teacher with the problem. It appears that ten year olds are supposed to monitor their learning, seek Extra Help, and self-advocate.

know your limits

glumbert - Women: Know your limits

so....thanks to Loyal Ktm Readers I now have TWO bad habits:

• Glumbert (Susan S)
• National Geographic Jigsaw Puzzle Generator (Anonymous)
  

South Park Mac vs PC

Boulder is a stoopid place...

I've just this moment noticed that Chris A has a blog!

control theory in a failing district...

5-star Comment left by a teacher (I assume this person is a teacher -- )

In control theory, there’s this wonderful treatment of a not so obvious reality. Without belaboring the math, it basically says that if you have a function that transforms something, you need to sample the output of the function and feedback an adjustment to the input in order to keep the whole system stable. Unstable systems fail (sometimes catastrophically).

So applying this theory to a classroom with a really good teacher you can picture something like; little empty minds in (the function input), lessons applied (the transformation), formative assessments (sampling), and individualized adjustments (the feedback loop) by the teacher.

My problem with NCLB is not with the testing but how the results are applied. Sticking with my analogy, the testing is nothing more than standardized (to a degree) sampling. The problem is that the measurement, the sample, is not applied to a feedback loop, its applied to the function. Results are not used to say, “Oh look, this child doesn’t understand common denominators yet. What adjustments can we make for this child?”

The results are used to take a big meat axe to the school district when a hundred tiny little scalpels would suffice. I work in a ‘failing’ district and I can tell you first hand that we are running an open loop control system that is wildly out of control. We have consultants pouring over us like hot fudge at an ice cream shop.

Their proscriptions always miss the mark and do not tweak the feedback loops. They are fussing about the scratches in the paint while the buildings burn. One of my favorite ‘solutions’ was the year of the bulletin board. We were, I kid you not, beat up for a year on the quality of our bulletin boards. We had bulletin board walkthroughs. We had bulletin boards so covered with crap that kids stopped looking at them. We had whole classroom blocks devoted to putting up a quality bulletin board.

Meanwhile, inside the classrooms, sixth graders that can’t add single digit numbers are being force fed a curriculum (the transform function) totally unsuited to where they are at by teachers who are not provided the sample data until 5 months after the testing for a class they no longer have.

It’s like trying to make course corrections for a rocket after it has crashed. It reminds of when I was a little kid playing at driving a car while sitting in a cardboard box holding an old bicycle wheel. The damn wheel never turned the box but it sure was fun!

I love this line:

We have consultants pouring over us like hot fudge at an ice cream shop.

Been there, done that!

This one's a keeper, too:

One of my favorite ‘solutions’ was the year of the bulletin board. We were, I kid you not, beat up for a year on the quality of our bulletin boards. We had bulletin board walkthroughs. We had bulletin boards so covered with crap that kids stopped looking at them. We had whole classroom blocks devoted to putting up a quality bulletin board.


I'm going to guess this person works in New York City.

Scratch that.

I'm going to hope this person works in New York City. I'd hate to think teachers elsewhere are having bulletin board walkthroughs.


update: Steve H on guess and check

Just saw Steve's riposte!

So, I guess a Bode Diagram is out of the question.

There is little feedback control in our schools. Testing results go straight onto report cards. It doesn't get fed back anywhere. That's because they think the input control is optimal already. Actually, they think the control starts in the home and report cards are the feedback to the parents. What the school does is part of the unchanging black box.

With NCLB, however, they are forced to apply a little bit of feedback to the system, but they can only use their favorite problem solving technique - guess and check.

Testing results go straight onto report cards....the control starts in the home and report cards are the feedback to the parents.

yup


see also:
The Logic of Failure by Dietrich Dorner

lgm on algebra in 8th grade

Comment left by lgm:

My college has shown statistics to interested alumni that say doing well in 8th grade algebra correlates with being successful in Calculus AB, which correlates to being most successful in earning an engineering degree in four years as opposed to dropping out or taking five/six years.

The "brain rule" that the brain needs 10 years to consolidate a memory comes as earth shattering news to me. Of course, I'd known about the 10-year rule for some time:

Some evidence that a great deal of practice, and not just talent, is a prerequisite for expertise is the "ten year rule," which states that individuals must practice intensively for at least 10 years before they are ready to make a substantive contribution to their field. What about prodigies like Mozart, who began composing at the age of six? Prodigies are very advanced for their age, but their contributions to their respective fields as children are widely considered to be ordinary. It is not until they are older (and have practiced more) that they achieve the works for which they are known.

Practice Makes Perfect -- But Only If You Practice Beyond the Point of Perfection
by Daniel Willingham
American Educator, Spring 2004

But it hadn't occurred to me that memory per se -- memory for ordinary, everyday skills and knowledge, as opposed to the memory involved in expert performance -- might also require 10 years to gel.

I am completely blown away by this.

Assuming John Medina is citing a field of research separate from the research on expertise (I'll find out when my copy of his book arrives), I'd say we have converging lines of evidence both for the 10-year rule and for assuming a causal rather than merely correlational relationship between algebra in the 8th grade, success in AP Calculus AB, and obtaining an engineering degree in four years as opposed to 5 or 6.

The fact that constructivist curricula are slower than the semi-traditional curricula they replaced is horrifying.

Wu's fractions: Common Denominators

This is the 3rd in a series of posts fleshing out the material written by Hung Hsi Wu in Critical Concepts for Understanding Fractions. See also Part I and Part II.

Last time, we left off after how to represent fractions on the number line. We showed that while a number is unique, and has a unique position on the number line, there were many different representations for that number based on how the number line was partitioned into pieces.

Recall we showed that 4/3 is the same as (5 x 4)/(5 x 3) with this example: locate 4/3 by its unique spot on the number line. We do this by breaking each unit into thirds, and then jumping 4 such 3rd-units to the right:


Now we partition each 3rd-unit into 5ths. Doing so immediately gives us 15-ths for each unit. Now we count how many 15ths our point 4/3 is: the answer is 20.



While our example was specific, our technique was not. Any fraction would have worked, and any new non zero partitioning. This demonstrated for us the mathematical fact: for all whole numbers k, m, and n, (such that n is not equal to zero and k is not equal to zero), m/n = km/kn.

In other words, m/n and km/kn are equivalent fractions. The equality symbol above tells us that these two quantities are the same. We can understand that to mean that there is one location on the number line that represents m/n, and it's the same location as that for km/kn, for any k, m, and n (such that n and k are nonzero.)

We note here that this is a good place to reinforce your comfort with commutativity of multiplication. We want students to feel comfortable recognizing that we could just as easily have said m/n = mk/nk and that would also be an equivalent fraction. This is where mastery of the underlying multiplication table is so important. The way to really believe the commutativity is to already know it's true for all of the natural numbers. We want to be able to multiply a fraction by k/k from either side without confusion.

This is also a good time to discuss a fairly beautiful fraction: k/k. Remember that our definition of a fraction is: the fraction m/n is the point on the number line, when we partition each unit into equal nths, and then make m jumps to the right of Zero on those new nth-hash marks. So the fraction k/k is the point on the number line when we partition each unit into equal kths and then make k jumps to the right of zero. This ALWAYS lands us at 1. So k/k is the same number as 1.

For students that are clicking this together, this is another way to see why m/n = mk/nk, but remember, we didn't resort to that explanation in the first place, because we'd like to become familiar with manipulating expressions like mk/nk WITHOUT reducing them. We are trying to build up more helpful denominators, to learn how to create new denominators, not just reduce them. Real mastery of k/k doesn't just lead us to say "that's 1, that cancels" but to say "we can replace 1 with k/k!"

Equivalent fractions are useful because they allow us to move quickly to compare fractions to each other, by use of this fact: Any two fractions may be represented as two fractions with the same denominator.

Mathematically, how would you do this? Well, take 2 fractions: m/n and k/l. m/n is the same as ml/nl. k/l is the same as nk/nl.
So now both m/n and k/l can be represented with the denominator nl.

Why do we care? Because now we can easily compare fractions, and we will now be better able to add and subtract them. Consider first the comparison. Two fractions with the same denominator can both be represented easily on the number line without confusion. Start with m/n and k/l. They become ml/nl and kn/nl respectively. ml/nl and kn/nl are represented by locations on the same sequence of (nl)ths. ml/nl is to the left of kn/nl if ml . By converting to a common denominator, we were able to solve the problem without having to graph out two different denominators. All we did was look at the numerators.

This leads to a specific method for comparing fractions. We convert each fraction to having the same denominator, and then look to see which numerator is bigger. In other words:

for all whole numbers k,l,m,n, k/l= m/n is equivalent to kn = ml .
This is just what we said above. It is also called the Cross Multiplication algorithm. (multiply the bottom of the left and the top of the right; multiply the bottom of the right and the top of the left. Which product is bigger?) In that case, we shorten the original procedure by ignoring that common denominator nl and just comparing the numerators. But indirectly, we are exploiting the fact that these two fractions are now represented by the same common denominator to determine which is larger or smaller.

Just as comparing two fractions is easier when you have common denominators, adding and subtracting them is easier too. We'll pick this up in the next post.

Update: above links fixed and should point correctly now!

Fixing NCLB

Jim Ryan has a piece posted at Slate on how to fix NCLB. It's worth reading in its entirety, but, not surprisingly, I disagree with some of his points. In the interest of space, I've paraphrased some of his key points; I hope I've been fair to his arguments, but, again, I would recommend going back to his original article.

Expand testing beyond Reading/Math/Science, to include history, literature, geography, art, and music. I think this is a terrible idea. I'm a huge advocate of standardized testing, but I actually favor scrapping science testing on a national level. Reading and math are special is because (1) they are fundamental to learning every other subject, and (2) they can be tested relatively cheaply and objectively compared to other subjects. On a more practical level, history, lit, art, and music curricula is decided at the locally - which is exactly as it should be. Testing for these subjects on a national level essentially establishes a national curriculum; if you thought standards are lax now, wait until you see the pablum produced by a national curriculum which caters to every interest group from every last corner of the country. And if you think you're annoyed by evolution/ID debates, just wait until we have to decide what goes into the history books.

Administer fewer tests. Ryan suggests testing only in 4th, 8th, and 11th grades. I think Ryan has it backwards; he wants comprehensive tests on many subjects every few years. Tests can be a great diagnostic tool, but they're only useful if you're getting data quickly and regularly. It makes much more sense to test a few subjects quickly, and often.

I think what we need are short, multiple choice tests on reading & math, given three times a year at each grade level. Each test should take no longer than 60 minutes each, and I like multiple-choice because they can be graded by machine very quickly. This means that a school should be able to administer, grade, and report the tests in a single day, allowing you to (a) schedule them on a Saturday to avoid interfering with the school year, and (b) get the results back almost immediately, allowing you to diagnose problem areas and adjust your teaching accordingly.

Rank schools; don't prescribe punishments. - I actually mostly agree with him on this one. Simply labelling a school as failing is the bureaucratic equivalent of giving a student an 'F' on his final without marking any of the wrong answers. Going back to my earlier point, what we need is a diagnostic allowing every school in the country to see what they are doing well, and what they are doing poorly. He also mentions another good idea: setting a uniform standards on measuring graduation rates.

Teachers and Money - This is the most curious passage. He specifically mentions low supplies of math/science teachers, and teachers in poor/minority districts, but offers a simplistic answer of 'more money'. He hints at it, but stops just short of what I believe is the full answer: 'more money, but not for everyone.' In other words, differential pay scales that reflect actual supply and demand for teachers. This is very likely to mean significantly higher pay for math/science teachers, and slightly lower pay for humanities teachers. Of course, I suspect that this is just the surface of the problem, which is exactly where he continues...

Teachers and Prestige. - Again, I think he gets it half-right when he says "respect, prestige, and decent working conditions also matter.", but I think he gets it wrong when he continues, "[the Federal government] can create a teaching program that restores prestige to the profession." Again, I think he gets it completely backwards - prestige is not something that can be created out of the ether, but something that has to be earned and developed from the ground up. This brings us back to a recurring topic here at KTM: the professionalization of teachers and teacher training. That is a subject far too deep to cover here, so I'll save that for a separate post (which I hope to complete sometime in this century).

Panel agnostic on what makes a good math teacher

Education Week weighs in on one of the math panel's findings.

Essential Qualities of Math Teaching Remain Unknown

In a stunning reversal of the oft-heard phrase "Research shows...", the panel is said to claim that research does not show what makes a good math teacher:

Research does not show conclusively which professional credentials demonstrate whether math teachers are effective in the classroom, the report found. It does not show what college math content and coursework are most essential for teachers. Nor does it show what kinds of preservice, professional-development, or alternative education programs best prepare them to teach.

As a result, while the report of the National Mathematics Advisory Panel, released last month, offers numerous conclusions about math curriculum, cognition, and instruction, many of its recommendations about improving teaching are more tentative and amount to a call for more research.

“It is, in some ways, where the action has to come next,” said Deborah Loewenberg Ball, the member of the panel who chaired its working group on teacher issues.

“We should put a lot of careful effort over the next decade into this issue so that we can be in a much different place 10 years from now.”

The uncertainty about math teaching skills emerges at a time when policymakers at all levels see a need to boost students’ math and science achievement as a key to sustaining the nation’s future economic health and producing a skilled workforce.

One reason the panel found a paucity of evidence on effective math instruction is that it set a high standard for the type of research it would accept, as Ms. Ball acknowledged.
Yet its members found a deeper pool of research in other areas of math, such as how students learn in the subject, and how students’ confidence in their ability influences their persistence and engagement in math study.

Without combing through the research, I can think of a few attributes that make a good math teacher. At the top of the list I would place mastery of topics to be taught and the ability to explain well.

Duh!

you can't cram math (or anything else)

Numerous parents here have spent years lobbying our high-performing, generously-funded district ($22,000 per pupil spending) to move to the international standard for math education. That being: algebra in the 8th grade.

We have approximately 30% of our 8th graders taking algebra. The figure at KIPP, in the Bronx, is 80%. ($10,000 per pupil spending, roughly)

They're not going to do it. They're so not going to do it they're not even going to say 'no.' They're just not going to do it.

While we're on the subject of well-funded school districts saying 'no,' I should add that the middle school is also not going to allow more students to take Earth Science in the 8th grade. Only forty-eight students, of 150 or so, currently take Earth Science, compared to 100% of students in Pelham. However, in the view of the school that is 48 students too many. As the chair of the science department told us, "If it were up to me, I wouldn't offer accelerated courses to any students in the middle school, but this community demands it."

The district argues, in meetings with parents, that learning depends upon maturity. Not all students are mature enough to learn Earth Science in the 8th grade. Or algebra.

Of course, maturity has nothing to do with ability to learn, as the National Math Advisory Panel reports. However, maturity has everything to do with a student being able to monitor his learning instead of depending on his teacher to perform this function. So, yes. It's easier to teach Earth Science to a high school sophomore than to a student in the 8th grade.

So why do parents continue to lobby school districts across the land to teach serious courses to younger kids?

What is the big deal, after all, about taking algebra in the 8th grade?

What's the difference when you take algebra so long as you get around to it sometime before college?


Brain Rules

It turns out there is a very good answer to that question.

It takes years to consolidate a memory. Not minutes, hours, or days but years. What you learn in first grade is not completely formed until your sophomore year in high school.

Rule # 6: Remember to repeat
John Medina

Bingo.

Here we have one of those facts of life many of us have picked up over the years but can neither verbalize in conversation with school officials nor defend as true, primarily because we don't realize we know it.

We don't know what we know. ^*

On the other hand, when we hear someone else verbalize it, we recognize it as true of our own experience. At least, I did, when I read this statement by James Milgram:

First of all, I claim that taking -- even asking to take it out of the curriculum -- shows a profound ignorance of the subject of mathematics. The point is, in mathematics, many, many skills develop over an extended period of time and are not really fully exploited until perhaps 10, 12, or even 15 years after they've been introduced. Some skills begin to develop in the first or second grade and they do not come to fruition or see their major applications until maybe the second year of college. This happens a lot in mathematics and long division is one of the key examples.

I'm going to guess that this is another reason why Singapore students are so far ahead of American students. Singapore students are doing simple algebra in the 5th grade. They're doing simple algebra in the 5th grade, and they're not dipping in and out of simple algebra, either; they're not being "exposed" to "algebraic thinking."

They're learning what they're learning to mastery.

At age 10.


^* not to be confused with known knowns, known unknowns, and unknown unknowns.

the future of America

from educationgadfly

don't try this at home

source:
Troika

how to...

...instantly transform a video billboard into a piece of public art.

First it was the math wars....

First it was the math wars...and now this.

As if getting featured in the New York Times and NBC Nightly News wasn't enough, now Ridgewood New Jersey has cars falling from the sky.

National Math Panel and methodology

Cross-posted from "After the Math Panel," a blog originating from Ridgewood New Jersey:

Instructional practices come and go, and some should flee faster than others.

The National Math Panel has scrutinized only the most rigorous studies to draw its conclusions. Not surprisingly, some of the panel's findings cast doubt on techniques recently in use--even in the best school districts.

Below are some interesting points extracted directly from the panel's final report. Administrators and teachers should take note of these, and consider them in light of current practices and future professional development. Schools of Education should also take a hard look.

The first list consists of direct quotes from the panel. The second list is a summary of this blogger's views and opinions, mapped to the first list. The final list highlights a few points of interest.

National Math Panel statements about Instructional Practices

(these are direct quotes)

1. Claims based on Piaget’s highly influential theory, and related theories of "developmental appropriateness" that children of particular ages cannot learn certain content because they are "too young," "not in the appropriate stage," or "not ready" have consistently been shown to be wrong. Nor are claims justified that children cannot learn particular ideas because their brains are insufficiently developed, even if they possess the prerequisite knowledge for learning the ideas.


2. The sociocultural perspective of Vygotsky has also been influential in education. It characterizes learning as a social induction process through which learners become increasingly independent through the tutelage of more knowledgeable peers and adults. However, its utility in mathematics classrooms and mathematics curricula remains to be scientifically tested.


3. The Panel recommends the scaling-up and experimental evaluation of support-focused interventions that have been shown to improve the mathematics outcomes of African-American and Hispanic students. [However,] average gender differences are small or nonexistent, and our society’s focus on them has diverted attention from the essential task of raising the scores of both boys and girls.


4. All-encompassing recommendations that instruction should be entirely "student centered" or "teacher directed" are not supported by research. If such recommendations exist, they should be rescinded. If they are being considered, they should be avoided. High-quality research does not support the exclusive use of either approach.


5. The Panel’s review of the literature addressed the question of whether using "real-world" contexts to introduce and teach mathematical topics and procedures is preferable to using more typical instructional approaches. For certain populations (upper elementary and middle grade students, and remedial ninth-graders) and for specific domains of mathematics (fraction computation, basic equation solving, and function representation), instruction that features the use of "real-world" contexts has a positive impact on certain types of problem solving. However, these results are not sufficient as a basis for widespread policy recommendations. Additional research is needed to explore the use of "real-world" problems in other mathematical domains, at other grade levels, and with varied definitions of "real-world" problems.


6. The Panel’s survey of the nation’s algebra teachers indicated that the use of calculators in prior grades was one of their concerns (National Mathematics Advisory Panel, 2008). The Panel cautions that to the degree that calculators impede the development of automaticity, fluency in computation will be adversely affected. The Panel recommends that high-quality research on particular uses of calculators be pursued, including both their short- and long-term effects on computation, problem solving, and conceptual understanding.


7. Research has been conducted on a variety of cooperative learning approaches. One such approach, Team Assisted Individualization (TAI), has been shown to improve students’ computation skills. This highly structured instructional approach involves heterogeneous groups of students helping each other, individualized problems based on student performance on a diagnostic test, specific teacher guidance, and rewards based on both group and individual performance. Effects of TAI on conceptual understanding and problem solving were not significant. There is suggestive evidence that peer tutoring improves computation skills in the elementary grades. However, additional research is needed.


8. Use of formative assessments in mathematics can lead to increased precision in how instructional time is used in class and can assist teachers in identifying specific instructional needs. Formative measures provide guidance as to the specific topics needed for assistance. Results [of studies] suggest that use of formative assessments benefited students at all ability levels. More studies are needed. Formative assessment should be an integral component of instructional practice in mathematics.

Summary:

1. Piaget's theories are not reliable for mathematics education. Interestingly, the constructivist approach to teaching is based on Piaget's theories. This finding of the panel casts grave doubt on the validity of a constructivist model for the teaching of mathematics.


2. The use of peer groups for the purpose of students teaching other students has never been tested, and therefore should be used sparingly and with caution.


3. Teaching methods specifically intended to reach girls should be dropped.


4. "Discovery" has always been a useful teaching approach and continues to be. The "discovery" approach can once again take its rightful place as one of many teaching techniques, rather than the dominant or only one, as it has in constructivist schools and classrooms.


5. The broad policy of using real-world problems to introduce and teach mathematical concepts has not been sufficiently tested, and should be restricted to upper grades, and then only to certain domains of mathematics.


6. The use of calculators before ninth grade has not only not been tested, the panel cautions that their use before grade nine interferes with the development of automaticity and fluency. Therefore, their use should be dropped until studies can be done.


7. Cooperative learning helps develop computation skills but not necessarily conceptual understanding or problem solving. Until further testing is done, cooperative learning should be limited to use for developing computation skills.


8. The increasing use of "formative assessment," also known as "authentic assessment" (assessment which is ongoing as opposed to traditional tests)is a good idea, and should continue.

A Few Points of Interest:

• It is noteworthy that while the panel was quite negative about early use of calculators, they were much more positive about the use of computer-assisted instruction. So much for educators lauding use of all technology. They need to think a little more critically about which technology.



• The practice of "formative assessment," a method used increasingly in some schools and often referred to as "authentic assessment," while understandably questioned by parents, has been tested and shows good results in the teaching of mathematics.



• Teachers and administrators should pursue practices that have been well-tested, and must exercise restraint with regard to practices that are not sufficiently tested. Parents, taxpayers, administrators, and teachers need to place their trust in science and an eclectic approach, rather than any one "ism."



• With regard to the evidence that cooperative learning can help develop
  computation skills, so can computer assistance. Either way, the student is prompted to focus on drill, and the teacher is freed up to work with other students. However, gifted literature is rife with anecdotes of negative impact on the student who is leaned on too much. It is wise to exercise caution, therefore, until studies of gifted students can be scrutinized more closely to determine the extent of negative impact.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Dana Huff: The Homework on Homework #2

Two weeks ago, Dana Huff summed up a discussion on homework. She comes back with round two: a discussion of Harris Cooper's 1989 article, “Synthesis of Research on Homework.”

One of the frustrations that Huff and I share is that there's no standard definition of homework. Cooper provides a tidy one:

Cooper defined homework as “tasks assigned to students by school teachers that are meant to be carried out during non-school hours”

What is your children's schools' definition of homework?

Previous related posts at Dana Huff's blog, Huffenglish.com: (I have elucidated some titles)
Grading to Communicate
Dana Huff, Agent Provocateur
Homework
Researching the Efficacy ofHomework Correction Techniques
Brief review of Classroom Instruction That Works (Marzano et al.)

Previous related posts here at KTM:
dy/dan on "why I don't assign homework"
Steve on teachers, homework, "Extra Help"
Steve on "collecting and correcting"
Effect size of collecting and correcting homework
All KTM posts labelled homework

citation for Cooper article:

Cooper, Harris. “Synthesis of Research on Homework.” Educational Leadership. 47.3 (November 1989): 85-91. Professional Development Collection. EBSCO. Weber School Library, Atlanta, GA. 29 March 2008. .